Cycles shadows different in live render and final render
This probably not a bug, but differences in settings between viewport/preview and the final render. 

Checklist:


Check that all objects that appear in the preview also appear in the final render, especially lamps. See: https://blender.stackexchange.com/a/1114/245 

Check that the sampling settings are the same for preview and final render. See https://blender.stackexchange.com/a/111591/245 for reference.
For completeness but probably not relevant: Check the modifiers for any difference between preview and final render.


Given your description there seems to be a problem in the rendering section so you might also use this answer for tracking down your problem. 

I specifically think that the amount of bounces is lower in the preview then in the final render. The increased diffuse lighting makes the shadow lighter.

EDIT:

The problem seems to be in the node setup:

The problem is, that the shadow has a strange mix between alpha and color:

The diameter of the shadow changes with the threshhold on the alpha channel!



Maybe you could try to use a shadowcatcher material/setup as described here. 

Investigating further, I think that you set up everything the right way, but this might be indeed a shortcoming of blender as discussed in the developemnt group here.